Math 211 Math 211 Lecture #18 Properties of Solution Spaces - - PowerPoint PPT Presentation

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Math 211 Math 211

Lecture #18 Properties of Solution Spaces February 26, 2001

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Method of Solution for Ax = b Method of Solution for Ax = b

• Use the augmented matrix M = [A, b].
• Eliminate as many coefficients as possible.

⋄ Use row operations to reduce to row echelon form.

• Write down the simplified system.
• Backsolve.

⋄ Assign arbitrary values to the free variables. ⋄ Solve for the pivot variables.

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Consistent Systems Consistent Systems

• A system is consistent if it has solutions.

⋄ The solution set is not the empty set.

• A system is consistent if and only if the simplified

version (after elimination) is consistent.

• This is true if and only if the last column (after

elimination) does not contain a pivot.

Method Consistent

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Examples Examples

A =

 

−3 6 −2 4 −1 2

 

b1 =

 

2 3 −5

 

b2 =

 

−9 −6 7

 

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Homogeneous Systems Homogeneous Systems

Example A =

 

−5 −4 −2 −6 −6 −2 30 27 11

  ⇒  

−5 −4 −2 −6 −6 −2 30 27 11

 

• During elimination the column of zeros is

unchanged.

• It is unnecessary to augment a homogeneous

system.

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Structure of the Solution Set Structure of the Solution Set

Theorem: Let xp be a particular solution to Axp = b.

• 1. If Axh = 0 then x = xp + xh also satisfies

Ax = b.

• 2. If Ax = b, then there is a vectorxh such that

Axh = 0 and x = xp + xh.

• Solution set for Ax = b is known if we know one

particular solution xp and the solution set for the homogeneous system Axh = 0.

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Square Matrices Square Matrices

• There are special kinds:

⋄ Singular and nonsingular. ⋄ Invertible and noninvertible.

• What do the terms mean?
• What are the relations between them?

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Singular and Nonsingular Matrices Singular and Nonsingular Matrices

The n × n matrix A is nonsingular if the equation Ax = b has a solution for any right hand side b. Proposition: The n × n matrix A is nonsingular if and only if the simplified matrix (after elimination) has only nonzero entries along the diagonal.

• In reduced row echelon form we get I.

Singular

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Examples Examples

A =

 

−17 −16 −6 18 18 6 6 3 3

 

A =

 

−17 −16 −6 18 18 6 6 3 4

 

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Proposition: If the n × n matrix A is nonsingular then the equation Ax = b has a unique solution for any right hand side b. Proposition: The n × n matrix A is singular if and only if the homogeneous equation Ax = 0 has a non-zero solution.

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Invertible Matrices Invertible Matrices

An n × n matrix A is invertible if there is an n × n matrix B such that AB = BA = I. The matrix B is called an inverse of A.

• If B1 and B2 are both inverses of A, then

B1 = B1(AB2) = (B1A)B2 = B2

• The inverse of A is denoted by A−1.
• Invertible ⇒ nonsingular .

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Computing the inverse A−1 Computing the inverse A−1

• Form the matrix [A, I].
• Do elimination until the matrix has the form

[I, B].

• Then A−1 = B.
• A matrix is invertible if and only if it is

nonsingular.

Solution set

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Solution Set of a Homogeneous System Solution Set of a Homogeneous System

Our goal is to understand such sets better. In particular we want to know:

• What are the properties of these solution sets?
• Is there a convenient way to describe them?